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Abelian Varieties: Geometry, Parameter Spaces, and Arithmetic

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Abelian Varieties: Geometry, Parameter Spaces, and Arithmetic ABELIAN VARIETIES: GEOMETRY, PARAMETER SPACES, AND ARITHMETIC BRIAN CONRAD In this article I wish to convey to beginners how analytic, algebro-geometric, and arithmetic techniques come together in the study of abelian varieties and their variation in families. For many modern arithmetic applications it is crucial that the entire theory admits an algebraic foundation (requiring the full force of the theory of schemes and beyond), but much of our geometric intuition and expectation is guided by the complex-analytic theory. It is for this reason that the first three sections §1-§3 are devoted to an overview of the analytic aspects of the theory, with an emphasis on those structures and examples that will be seen to admit natural analogues in the algebraic theory. With this experience behind us, §4-5 explain how to make the appropriate definitions in the algebraic setting and what sorts of results and interesting examples one obtains. The final two sections (§6 and §7) are devoted to the study of analytic and algebraic families of abelian varieties parameterized by some “modular” varieties, with the aim of explaining (via the Main Theorem of Complex Multiplication) how these varieties fit within the framework of Shimura varieties. It is assumed that the reader has some prior exposure to the basics of algebraic geometry over an alge- braically closed field, and notions such as complex manifold and vector bundle over a manifold (as well as bundle operations, such as dual, tensor product, and pullback). We will certainly have to allow ourselves to work with algebro-geometric objects over a field that is not algebraically closed (so the reader unfamiliar with such things will have to take a lot on faith), and the proofs of many of the algebraic theorems that we state without proof require a solid command of the theory of schemes. Hence, these notes should be understood to be merely a survey of important notions, examples, and results for a reader who is taking their first steps into this vast and beautiful subject. Notation. Throughout, C denotes an algebraic closure of R that is fixed for all time. We write Z(1) to × denote√ the kernel of the exponential map exp : C √ C , so this is a free Z-module of rank 1 generated by ±2π −1; of course, there is no canonical choice of −1 in C. For any Z-module M we write M(1) to denote M ⊗Z Z(1). For example, R(1) is the “imaginary axis” in C (i.e., the −1-eigenspace for complex conjugation on the R-vector space C), C(1) is canonically isomorphic to C via the multiplication map C ⊗Z Z(1) → C, and for any positive integer N the function e(·)/N identifies (Z/NZ)(1) = Z(1)/N · Z(1) with the group µ (C) of Nth roots of unity in C. (Thus, for any prime ` we have that Z (1) ' lim Z(1)/`n · Z(1) is N ` ←− ` isomorphic to lim µ n (C) via the transition maps µ n+1 (C) µ n (C) defined by z 7→ z .) ←− ` ` ` √ By working systematically with Z(1) we can avoid using a choice of −1 throughout the analytic theory (as we must be able to do if we are to give definitions that admit algebraic analogues, since the algebraic theory√ does not know the concept of orientation for the complex plane that is equivalent to making a choice of −1). In this spirit, for any z ∈ C we write Re(z) to denote the “real part” (z + z)/2 = (1/2)TrC/R(z) and zim to denote the “imaginary component” (z − z)/2 = z − Re(z); these are canonical, whereas the classical “imaginary part” Im(z) is not. If M is a finite and free module over a commutative ring R (especially Z or a field) then we write M ∨ to ∨ denote the dual module HomR(M, R). For example, if M is a finite free Z-module then M (1) is naturally identified with the module Hom(M, Z(1)) of Z(1)-valued Z-linear forms on M. Date: July 8, 2005. 1991 Mathematics Subject Classification. Primary 14G22; Secondary 14H52. This work was partially supported by grants from the NSF and the Alfred P. Sloan Foundation. I would like to thank the organizers of the summer school for the invitation to present an introductory overview of this subject. 1 2 BRIAN CONRAD Contents 1. Basic analytic definitions and examples 2 1.1. Uniformizations 2 1.2. Endomorphisms and isogenies 7 1.3. CM tori 9 2. Analytic aspects of line bundles, duality, and pairings 13 2.1. The dual torus 14 2.2. Classification of line bundles 15 2.3. Correspondences and universal properties 19 2.4. Pairings and torsion 21 3. Analytic aspects of polarizations and CM tori 23 3.1. Polarizations 23 3.2. Siegel’s construction 27 3.3. Endomorphisms and polarizations 28 3.4. Tate modules 30 4. Basic algebraic definitions and examples 32 4.1. Maps and torsion 33 4.2. Tate modules and applications 35 4.3. Duality and Weil pairing 38 5. Algebraic theory of polarizations and endomorphisms 40 5.1. The Mumford construction 40 5.2. `-adic Riemann forms and the Rosati involution 44 6. Analytic models and algebraic descent 46 6.1. Analytic and algebraic modular curves 46 6.2. Characterization of algebraic models via Galois action 51 6.3. Heegner points and classical CM theory 60 6.4. Applications of CM theory to algebraicity 62 7. Siegel moduli spaces as Shimura varieties 65 7.1. Analytic Siegel families and moduli spaces 65 7.2. Siegel half-spaces as coset spaces 69 7.3. Algebraic Siegel modular varieties 72 7.4. Galois action at CM points 76 References 78 1. Basic analytic definitions and examples 1.1. Uniformizations. The beginning of the theory is: Definition 1.1.1. A complex torus is a compact connected complex Lie group. In the 1-dimensional case it is called an elliptic curve. Example 1.1.2. Let V be a finite-dimensional C-vector space with dimension g ≥ 1 and let Λ be a lattice in V (i.e., a co-compact discrete subgroup of V ). Equivalently, Λ is a discrete subgroup of V that is finite and free of rank 2g as a Z-module, or in yet another formulation it is a finite free Z-submodule of V such that the induced R-linear map R ⊗Z Λ → V is an isomorphism. The quotient V/Λ has a natural structure of compact Hausdorff commutative topological group and it admits a unique structure of complex manifold with respect to which the projection map V → V/Λ is a local analytic isomorphism. As such, V/Λ is a complex torus. Remark 1.1.3. Beware that although the natural R-linear map R ⊗Z Λ → V in Example 1.1.2 is an iso- morphism, so a Z-basis of Λ provides an R-basis of V giving an R-linear identification V ' R2g carrying ABELIAN VARIETIES: GEOMETRY, PARAMETER SPACES, AND ARITHMETIC 3 Λ over to Z2g, there is generally no C-basis of V with respect to which the description of Λ in V is equally straightforward. In other words, the C-structure put on the R-vector space R ⊗Z Λ via its identification with V is a subtle structure. However, it can be described in somewhat concrete terms if we choose suitable bases. For example, in the classical case g = 1 we may pick a basis {λ, λ0} of Λ and use λ0 as a basis of V , so the resulting identification of V with C carries Λ to the lattice Zτ ⊕ Z where τ ∈ C − R satisfies λ = τλ0 in V . Conversely, for any τ ∈ C − R certainly Zτ ⊕ Z is a lattice in C. Hence, in the case g = 1 we may describe all lattice embeddings Λ ,→ V in the concrete form Zτ ⊕ Z ⊆ C; the choice of τ is not intrinsic to the lattice (and, following Deligne, it determines an orientation of the R-vector space C via the nonzero 2 2 element τ ∧ 1 ∈ ∧R(R ⊗Z Λ) = ∧R(C); this is opposite to the “classical” orientation). Example 1.1.4. Let Λ ⊆ C be a lattice. In the classical theory of elliptic functions it is proved that the Weierstrass ℘-function 1 X 1 1 ℘ (z) = + − Λ z2 (z − λ)2 λ2 λ∈Λ−{0} is uniformly convergent on compacts in C − Λ and is meromorphic in C with Λ-periodicity and double poles along Λ. For the constants X 1 X 1 g (Λ) = 60 , g (Λ) = 140 2 λ4 3 λ6 λ∈Λ−{0} λ∈Λ−{0} 0 2 3 one has the non-linear differential equation ℘Λ = 4℘Λ −g2(Λ)℘Λ −g3(Λ) for which the cubic has discriminant 3 2 2 2 ∆(Λ) = g2(Λ) − 27g3(Λ) that is non-zero. Hence, the projective curve E in CP with affine model y = 3 0 4x −g2(Λ)x−g3(Λ) is smooth with [0, 1, 0] as its unique point on the line at infinity and z 7→ (℘Λ(z), ℘Λ(z)) uniquely extends to an analytic isomorphism C/Λ ' E carrying 0 to [0, 1, 0]. Moreover, the ratio 1728g (Λ)3 j(Λ) = 2 ∆(Λ) is called the j-invariant of C/Λ and it determines the isomorphism class of the elliptic curve C/Λ. Conversely, every smooth plane cubic y3 = 4x3 − ax2 − b with a3 − 27b2 6= 0 arises from some Λ as above.
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